Stationary covariance bound for reversible Markov chains

If $\Vert P{\Vert }_{L_0^2(\mu )}\le r<1$, a stationary reversible Markov chain satisfiesConsequently the variance of a sum of $N$ consecutive values is at most $N(1+r)/(1-r)$ times the stationary variance of one value. This bounds the Markov chain Monte Carlo asymptotic variance uniformly over square-integrable functions.